3.377 \(\int \frac{1}{x^7 (1+3 x^4+x^8)} \, dx\)

Optimal. Leaf size=97 \[ \frac{3}{2 x^2}-\frac{1}{6 x^6}-\frac{1}{2} \sqrt{\frac{1}{10} \left (123-55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{10} \left (123+55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

[Out]

-1/(6*x^6) + 3/(2*x^2) - (Sqrt[(123 - 55*Sqrt[5])/10]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(123 + 55*S
qrt[5])/10]*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

________________________________________________________________________________________

Rubi [A]  time = 0.134403, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1359, 1123, 1281, 1166, 203} \[ \frac{3}{2 x^2}-\frac{1}{6 x^6}-\frac{1}{2} \sqrt{\frac{1}{10} \left (123-55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{10} \left (123+55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 + 3*x^4 + x^8)),x]

[Out]

-1/(6*x^6) + 3/(2*x^2) - (Sqrt[(123 - 55*Sqrt[5])/10]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(123 + 55*S
qrt[5])/10]*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

Rule 1359

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[
1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k) + c*x^((2*n)/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b,
c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 1123

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2 +
c*x^4)^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p +
5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && In
tegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*(
f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^7 \left (1+3 x^4+x^8\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+3 x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-9-3 x^2}{x^2 \left (1+3 x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{3}{2 x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{-24-9 x^2}{1+3 x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{3}{2 x^2}-\frac{1}{20} \left (-15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )+\frac{1}{20} \left (15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{3}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{10} \left (123-55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{20} \sqrt{1230+550 \sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.0194256, size = 73, normalized size = 0.75 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{3 \text{$\#$1}^4 \log (x-\text{$\#$1})+8 \log (x-\text{$\#$1})}{2 \text{$\#$1}^6+3 \text{$\#$1}^2}\& \right ]+\frac{3}{2 x^2}-\frac{1}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 + 3*x^4 + x^8)),x]

[Out]

-1/(6*x^6) + 3/(2*x^2) + RootSum[1 + 3*#1^4 + #1^8 & , (8*Log[x - #1] + 3*Log[x - #1]*#1^4)/(3*#1^2 + 2*#1^6)
& ]/4

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 122, normalized size = 1.3 \begin{align*}{\frac{7\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }+3\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-{\frac{7\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2+2\,\sqrt{5}}} \right ) }+3\,{\frac{1}{2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2+2\,\sqrt{5}}} \right ) }-{\frac{1}{6\,{x}^{6}}}+{\frac{3}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^8+3*x^4+1),x)

[Out]

7/5*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))+3/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))-7/5*5^(1
/2)/(2+2*5^(1/2))*arctan(4*x^2/(2+2*5^(1/2)))+3/(2+2*5^(1/2))*arctan(4*x^2/(2+2*5^(1/2)))-1/6/x^6+3/2/x^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{9 \, x^{4} - 1}{6 \, x^{6}} + \int \frac{{\left (3 \, x^{4} + 8\right )} x}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

1/6*(9*x^4 - 1)/x^6 + integrate((3*x^4 + 8)*x/(x^8 + 3*x^4 + 1), x)

________________________________________________________________________________________

Fricas [B]  time = 1.77081, size = 564, normalized size = 5.81 \begin{align*} \frac{3 \, \sqrt{10} x^{6} \sqrt{-55 \, \sqrt{5} + 123} \arctan \left (\frac{1}{40} \, \sqrt{10} \sqrt{2 \, x^{4} + \sqrt{5} + 3}{\left (7 \, \sqrt{5} \sqrt{2} + 15 \, \sqrt{2}\right )} \sqrt{-55 \, \sqrt{5} + 123} - \frac{1}{20} \, \sqrt{10}{\left (7 \, \sqrt{5} x^{2} + 15 \, x^{2}\right )} \sqrt{-55 \, \sqrt{5} + 123}\right ) - 3 \, \sqrt{10} x^{6} \sqrt{55 \, \sqrt{5} + 123} \arctan \left (\frac{1}{40} \,{\left (\sqrt{10} \sqrt{2 \, x^{4} - \sqrt{5} + 3}{\left (7 \, \sqrt{5} \sqrt{2} - 15 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (7 \, \sqrt{5} x^{2} - 15 \, x^{2}\right )}\right )} \sqrt{55 \, \sqrt{5} + 123}\right ) + 45 \, x^{4} - 5}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

1/30*(3*sqrt(10)*x^6*sqrt(-55*sqrt(5) + 123)*arctan(1/40*sqrt(10)*sqrt(2*x^4 + sqrt(5) + 3)*(7*sqrt(5)*sqrt(2)
+ 15*sqrt(2))*sqrt(-55*sqrt(5) + 123) - 1/20*sqrt(10)*(7*sqrt(5)*x^2 + 15*x^2)*sqrt(-55*sqrt(5) + 123)) - 3*s
qrt(10)*x^6*sqrt(55*sqrt(5) + 123)*arctan(1/40*(sqrt(10)*sqrt(2*x^4 - sqrt(5) + 3)*(7*sqrt(5)*sqrt(2) - 15*sqr
t(2)) - 2*sqrt(10)*(7*sqrt(5)*x^2 - 15*x^2))*sqrt(55*sqrt(5) + 123)) + 45*x^4 - 5)/x^6

________________________________________________________________________________________

Sympy [A]  time = 0.274987, size = 65, normalized size = 0.67 \begin{align*} 2 \left (\frac{11 \sqrt{5}}{40} + \frac{5}{8}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{-1 + \sqrt{5}} \right )} - 2 \left (\frac{5}{8} - \frac{11 \sqrt{5}}{40}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{1 + \sqrt{5}} \right )} + \frac{9 x^{4} - 1}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**8+3*x**4+1),x)

[Out]

2*(11*sqrt(5)/40 + 5/8)*atan(2*x**2/(-1 + sqrt(5))) - 2*(5/8 - 11*sqrt(5)/40)*atan(2*x**2/(1 + sqrt(5))) + (9*
x**4 - 1)/(6*x**6)

________________________________________________________________________________________

Giac [A]  time = 1.24186, size = 104, normalized size = 1.07 \begin{align*} \frac{1}{20} \,{\left (3 \, x^{4}{\left (\sqrt{5} - 5\right )} + 8 \, \sqrt{5} - 40\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} + 1}\right ) + \frac{1}{20} \,{\left (3 \, x^{4}{\left (\sqrt{5} + 5\right )} + 8 \, \sqrt{5} + 40\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} - 1}\right ) + \frac{9 \, x^{4} - 1}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

1/20*(3*x^4*(sqrt(5) - 5) + 8*sqrt(5) - 40)*arctan(2*x^2/(sqrt(5) + 1)) + 1/20*(3*x^4*(sqrt(5) + 5) + 8*sqrt(5
) + 40)*arctan(2*x^2/(sqrt(5) - 1)) + 1/6*(9*x^4 - 1)/x^6